From differential crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to differential crossed modules. Gauging procedures in supergravity rely on a pairing -- the embedding tensor -- between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which differential crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra -- its associated tensor hierarchy -- whose restriction to the category of differential crossed modules is the canonical assignment associating to any differential crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of differential crossed modules and that their associated tensor hierarchies can be considered as some kind of 'lie-ization' of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.Comment: v4, 46 pages, slight improvement in main theorem, examples adde